0

The thief is at point $(0,1)$. Before he starts to move, the police bureau of the $\mathbb{R^2}$ plain can freely place countably infinite officers along the x-axis. We know that

  • The thief and the officers move simultaneously and continuously, their maximal speeds being $V_t$ and $V_o$ respectively.
  • The officers are restricted to move along the x-axis. They're ghostlike and pass right through each other without collision.
  • The thief and the officers are points. The thief is caught if his coordinates coincide with those of an officer.

Can the thief cross over the x-axis to the negative y hemisphere without getting caught? The answer is obvious no if $V_t\leq V_o$. But can he if $V_t\gt V_o$?

What happens if instead of occupying the x-axis, the officers occupy a circle around the thief? Can the thief break out of the circle?

Eric
  • 1,909
  • Since $\mathbb Q$ is countable and dense in $\mathbb R$, every time the thief starts to move to a new real $x$-coordinate, there exists a rational officer that is arbitrarily close to it. In other words, an officer can always get there in arbitrarily small amount of time? – Vepir Oct 01 '21 at 12:38
  • @Vepir Yes, they can. – Eric Oct 01 '21 at 14:38
  • Then how can thief ever escape if there is always an officer in front of it, regardless of speeds? – Vepir Oct 01 '21 at 15:14
  • @Vepir I have no idea whether he can or can not. But what's your officers' plan to make sure the thief is caught if you think he can't make it? How do they move? – Eric Oct 01 '21 at 15:51
  • @Vepir Let $Q(t)$ be the set of points occupied by the officers, $P(t)$ the set of points on the x-axis that are not occupied by an officer, $S(t)$ the trajectory of the thief. If $S(t)\cap P(t)\neq \emptyset$ for some $t$, then the thief escapes. – Eric Oct 01 '21 at 16:16
  • I was saying that there is always (at every moment $t$) an officer standing on the same $x$-coordinate as the thief (all officers shift left or right in unison), because speeds do not matter as distances needed to be travelled by officers are some arbitrarily small $\epsilon\gt 0$. Therefore, the moment thief steps on the $x$-axis, it is caught. – Vepir Oct 01 '21 at 17:18
  • @Vepir I posted a more general version here https://mathoverflow.net/questions/405526/can-the-thief-escape – Eric Oct 05 '21 at 16:00

0 Answers0